Department of

November 2005 December 2005 January 2006 Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa 1 2 3 4 5 1 2 3 1 2 3 4 5 6 7 6 7 8 9 10 11 12 4 5 6 7 8 9 10 8 9 10 11 12 13 14 13 14 15 16 17 18 19 11 12 13 14 15 16 17 15 16 17 18 19 20 21 20 21 22 23 24 25 26 18 19 20 21 22 23 24 22 23 24 25 26 27 28 27 28 29 30 25 26 27 28 29 30 31 29 30 31

Friday, December 2, 2005

**Abstract:** Flat $G$-bundles on Riemannian manifold M are parametrised by representations of the fundamental group $\pi_1(M)$ into $G$. I will explain that for a special subset of the space of representations of a surface group into the symplectic group $Sp(2n,R)$ we can construct a splitting into two (non-flat) Lagrangian subbundles. Contraction and expanding properties of these subbundles imply that the holonomy representations of these symplectic vector bundles are quasiisometric embeddings, in particular they are faithful with discrete image. These special subsets of the space of representations can be viewed as generalized Teichm\"uller space. If time permits I will explain some relations with other moduli spaces.